The hyperbolic cover of an elliptic Weyl group

TL;DR

This paper explores hyperbolic covers of elliptic Weyl groups, establishing their isomorphisms, and applies these results to understand the structure of subcategories in weighted projective lines of tubular type.

Contribution

It demonstrates the isomorphism of hyperbolic covers with extended Coxeter systems and links braid group actions to subcategory posets in algebraic geometry.

Findings

Hyperbolic covers are isomorphic to extended Coxeter systems of star type.

Braid group actions are transitive on reduced reflection factorizations.

Poset of subcategories corresponds to an interval in the absolute order.

Abstract

In this paper, we study in detail the hyperbolic covers $\tilde{W}$ and $\hat{W}$ of an elliptic Weyl system introduced by Saito. We show that they are isomorphic and also isomorphic to an extended Coxeter system of star type. For $\tilde{c}$ a Coxeter transformation in $\tilde{W}$ we can conclude the Hurwitz transitivity of the braid group action on the set of reduced reflection factorizations of $\tilde{c}$ from the Hurwitz transitivity in extended Coxeter systems of star type. This then enables us to establish for a weighted projective line $\mathbb{X}$ of tubular type an order preserving bijection between the poset of thick subcategories of $\mathrm{coh}(\mathbb{X})$ generated by an exceptional sequence and the poset $[\mathrm{id}, \tilde{c}]$ ordered by the absolute order. In an Appendix, we study the hyperbolic cover of a Coxeter system.